Two Theories of Special Relativity ?

TL;DR

This paper explores two possible foundational theories of Special Relativity, one based on the Principle of Transformation Increment Ratio Limitation (PTIRL) and the other rejecting it, which could lead to different physical implications and experimental accessibility.

Contribution

It introduces PTIRL as a new physical axiom and discusses how its acceptance or rejection results in two distinct theories of Special Relativity.

Findings

PTIRL is closely related to boundedness conditions on space-time transformations.

Rejecting PTIRL leads to effects involving unlimited space/time intervals.

Two different theories of Special Relativity emerge depending on PTIRL acceptance.

Abstract

Recently, [3], it was shown that Special Relativity is in fact based on one single physical axiom which is that of Reciprocity. Originally, Einstein, [1], established Special Relativity on two physical axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Soon after, [4,5], it was shown that the Galilean Relativity alone is sufficient for Special Relativity. Here it is important to note that, implicitly, three more assumptions have been used on space-time coordinate transformations, namely, the homogeneity of space-time, the isotropy of space, and a mathematical condition of smoothness type. In [3], a boundedness condition on space-time coordinate transformations is used instead of a usual mathematical smoothness type condition. In this paper it is shown that the respective boundedness condition is closely related to a Principle of Transformation Increment Ratio Limitation, or in short, PTIRL, which has an obvious physical meaning. It is also shown that PTIRL is {\it not} a stronger assumption than that of the mentioned boundedness in [3]. Of interest is the fact that, by formulating PTIRL as a physical axiom, the possibility is opened up for the acceptance, or on the contrary, rejection of this physical axiom PTIRL, thus leading to {\it two} possible theories of Special Relativity. And to add further likelihood to such a possibility, the rejection of PTIRL leads easily to effects which involve unlimited time and/or space intervals, thus are not accessible to usual experimentation for the verification of their validity, or otherwise.